A269939 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-3 - 2\,x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||
| 2 | 1 | 1 | ||||||
| 3 | 3 | 1 | 4 | |||||
| 4 | 15 | 10 | 1 | 26 | ||||
| 5 | 105 | 105 | 25 | 1 | 236 | |||
| 6 | 945 | 1260 | 490 | 56 | 1 | 2752 | ||
| 7 | 10395 | 17325 | 9450 | 1918 | 119 | 1 | 39208 | |
| 8 | 135135 | 270270 | 190575 | 56980 | 6825 | 246 | 1 | 660032 |
Compute more
?
Technical details
- Node id
n-4f3d85d3aa6073ea· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-014ae34d4ff839c2
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 1; 3: 3, 1; 4: 15, 10, 1; 5: 105, 105, 25, 1; 6: 945, 1260, 490, 56, 1; 7: 10395, 17325, 9450, 1918, 119, 1; 8: 135135, 270270, 190575, 56980, 6825, 246, 1; 9: 2027025, 4729725, 4099095, 1636635, 302995, 22935, 501, 1; 10: 34459425, 91891800, 94594500, 47507460, 12122110, 1487200, 74316, 1012, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (2*n-3+(n-2)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 3 + (n - 2)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n - 3 + (n - 2)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
- Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m),…?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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from
Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m),…?
Related ?
Same fingerprint ?
- A163936 → x ↦ x+1
- A008517 → x ↦ x/(x+1)
- A269939: Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n…
- A163936 → reverse → x ↦ x/(x+1)
- A112007 → x ↦ x+1
- A134991: Triangle of Ward numbers T(n,k) read by rows.
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.